Inverse semigroup

The inverse semigroup is a term from the mathematical branch of algebra . He generalizes that of the group . Here, inverse elements are defined without reference to a neutral element .

definition

An inverse semigroup is a semigroup with the property that for each there is a clearly defined , inverse (in contrast to the inverse element related to a neutral element also relative inverse ) of called, with ${\displaystyle (A,\cdot )}$ ${\displaystyle x\in A}$ ${\displaystyle y\in A}$ ${\displaystyle x}$

${\displaystyle x\cdot y\cdot x=x}$and . ${\displaystyle y\cdot x\cdot y=y}$

Equivalent Definitions

With operation symbol

A semigroup is an inverse semigroup if idempotent elements commute and another operation is such that for all true ${\displaystyle (A,\cdot )}$ ${\displaystyle (-)^{*}\colon A\to A}$ ${\displaystyle x\in A}$

${\displaystyle x\cdot x^{*}\cdot x=x}$and . ${\displaystyle x^{*}\cdot x\cdot x^{*}=x^{*}}$

Purely algebraic

A semigroup is an inverse semigroup if there is another operation and the following equations are true for all of them: ${\displaystyle (A,\cdot )}$ ${\displaystyle (-)^{*}\colon A\to A}$ ${\displaystyle x,y\in A}$

• ${\displaystyle (x^{*})^{*}=x,}$
• ${\displaystyle x\cdot x^{*}\cdot x=x,}$
• ${\displaystyle x\cdot x^{*}\cdot y\cdot y^{*}=y\cdot y^{*}\cdot x\cdot x^{*}.}$

Examples and Applications

Each group is an inverse semigroup, with . ${\displaystyle x^{*}:=x^{-1}}$

Each semi-lattice is an inverse semi-group, with . ${\displaystyle x^{*}:=x}$

The definition of a "Meadow" is obtained by modifying the definition of a body as a special unitary commutative ring : instead of also requiring that a group is, it is required that an inverse semigroup is. The consequence is that “Meadows” can be axiomatized purely algebraically. The "division", defined as multiplication by the inverse, becomes total; it is . ${\displaystyle (R,+,\cdot )}$ ${\displaystyle (R\setminus \{0\},\cdot )}$ ${\displaystyle (R,\cdot )}$ ${\displaystyle 1/0=0}$

properties

For every element of an inverse semigroup is always idempotent. In addition, every idempotent element can be represented in this form, since . ${\displaystyle x}$ ${\displaystyle x^{*}\cdot x}$ ${\displaystyle e}$ ${\displaystyle e=e^{*}\cdot e}$

As in groups is and . ${\displaystyle (x^{*})^{*}=x}$ ${\displaystyle (x\cdot y)^{*}=y^{*}\cdot x^{*}}$

literature

• Alan Paterson: Groupoids, Inverse Semigroups, and their Operator Algebras . 1999, ISBN 0-8176-4051-7 .
• John Mackintosh Howie : Fundamentals of Semigroup Theory . 1995, ISBN 0-19-851194-9 .

Individual evidence

1. ↑ AH Clifford: Semigroups admitting relative inverses (=  Ann. Of Math. No. 42 ). 1941, p. 1037-1049 .
2. ↑ Alan Paterson: Groupoids, Inverse Semi Groups, and Their Operator Algebras . 1999, ISBN 0-8176-4051-7 , pp. 21 .
3. ^ Inversion semi-group . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
4. ^ JA Bergstra, Y. Hirshfeld, JV Tucker: Meadows and the equational specification of division . January 7, 2009, arxiv : 0901.0823 .